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To solve this problem, we need to find the first five terms of a geometric sequence. We are given the sixth term \( a_6 = 25 \) and the eighth term \( a_8 = 6.25 \). The formula for the \( n \)-th term of a geometric sequence is:

\[
a_n = a_1 \cdot r^{n-1}
\]

Where:
- \( a_n \) is the \( n \)-th term.
- \( a_1 \) is the first term.
- \( r \) is the common ratio.

### Step 1: Set up two equations for \( a_6 \) and \( a_8 \)

From the general formula, we have:

\[
a_6 = a_1 \cdot r^5 = 25
\]
\[
a_8 = a_1 \cdot r^7 = 6.25
\]

### Step 2: Solve for the common ratio \( r \)

To eliminate \( a_1 \), divide the second equation by the first:

\[
\frac{a_8}{a_6} = \frac{a_1 \cdot r^7}{a_1 \cdot r^5} = r^2
\]

This simplifies to:

\[
\frac{6.25}{25} = r^2
\]
\[
r^2 = 0.25
\]
\[
r = 0.5
\]

### Step 3: Solve for \( a_1 \)

Now substitute \( r = 0.5 \) into the equation for \( a_6 \):

\[
a_1 \cdot (0.5)^5 = 25
\]
\[
a_1 \cdot \frac{1}{32} = 25
\]
\[
a_1 = 25 \cdot 32 = 800
\]

### Step 4: Find the first five terms

Now that we know \( a_1 = 800 \) and \( r = 0.5 \), we can calculate the first five terms:

\[
a_1 = 800
\]
\[
a_2 = a_1 \cdot r = 800 \cdot 0.5 = 400
\]
\[
a_3 = a_2 \cdot r = 400 \cdot 0.5 = 200
\]
\[
a_4 = a_3 \cdot r = 200 \cdot 0.5 = 100
\]
\[
a_5 = a_4 \cdot r = 100 \cdot 0.5 = 50
\]

### Final Answer:
The first five terms of the geometric sequence are:
\[
a_1 = 800, \quad a_2 = 400, \quad a_3 = 200, \quad a_4 = 100, \quad a_5 = 50
\]

Let me know if you would like further details or clarifications!

### Related Questions:
1. How do you derive the formula for the \( n \)-th term of a geometric sequence?
2. What happens to the terms of a geometric sequence if the common ratio is greater than 1?
3. How would you solve the problem if you were given different terms of the sequence, say \( a_3 \) and \( a_7 \)?
4. Can you explain how geometric sequences are used in financial contexts, like compound interest?
5. What is the sum of the first five terms of this geometric sequence?

**Tip:** In geometric sequences, the ratio between successive terms is always constant, making it easier to detect and calculate unknown terms.

Write the first five terms of the geometric sequence, given any two terms: a_6 = 25, a_8 = 6.25.

Learn how to find the first five terms of a geometric sequence given two specific terms, a_6 = 25 and a_8 = 6.25. This problem covers key concepts in algebra and geometric sequences, suitable for students in Grades 9-12.

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